Definite Integrals and Their Properties

Every integral so far in this chapter has been an INDEFINITE integral — a family of functions, ending in $+C$. A definite integral instead carries two limits and evaluates to a single NUMBER, with no constant of integration left in the final answer.

Notation

The definite integral of $f(x)$ from $x=a$ to $x=b$ is written

where $a$ is the lower limit and $b$ is the upper limit.

The Fundamental Theorem of Integral Calculus

Why the constant of integration disappears: if $F(x)+C$ is used instead of $F(x)$, the constant cancels automatically — $\big[F(x)+C\big]_{a}^{b} = \big(F(b)+C\big)-\big(F(a)+C\big) = F(b)-F(a)$ — which is exactly why $C$ is never written when evaluating a definite integral.

Three properties used throughout this chapter

1. Reversing the limits changes the sign: $\displaystyle\int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx$.
2. Equal limits give zero: $\displaystyle\int_{a}^{a} f(x)\,dx = 0$ (there is no interval to accumulate over).
3. Additivity over sub-intervals: for any point $c$ between $a$ and $b$, $\displaystyle\int_{a}^{b} f(x)\,dx = \int_{a}^{c} f(x)\,dx + \int_{c}^{b} f(x)\,dx$ — the integral over the whole interval equals the sum of the integrals over the pieces.

Worked pattern …